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The colour of the Sun is white
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this post was submitted on 01 Oct 2023
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Am I missing something, or is the point about sunlight's spectral peak being different in frequency space than wavelength space non-sensical?
Wavelength and frequency are simply the inverse of each other, and two different ways of describing the same color. The color with peak intensity is what it is, whether you describe it unsing wavelength or frequency.
Maybe the author meant some kind of integral / area under the curve concept, since the shape of the curve is different using wavelength vs frequency as the x axis, but even then, the actual power output across some spectrum range is independent of whether you define the range in terms of wavelength or frequency.
https://www.scielo.br/j/rbef/a/mYqvM4Qc3KLmmfFRqMbCzhB/?lang=en
This is something that bothered me when I was in undergrad but now I've come to understand. The article above goes through the math of computing different Wien peaks for different representations of the spectral energy density.
In short, the Wien peaks are different because what the density function measures in a given parametrization is different. In frequency space the function measures the energy radiated in a small interval [f, f + df] while in wavelength space it measure the energy radiated in an interval [位, 位 + d位]. The function in these spaces will be different to account for the different amounts of energy radiated in these intervals, and as such the peaks are different too.
(I typed this on a phone kinda rushed so I could clarify it if you'd like)
Hey thanks for the reply! I'll admit that paper lost me pretty quickly, so I am probably missing a subtle point. But it feels deeply unintuitive since frequency and wavelength are just two different ways of describing the same physical quantity.
So if I have a given source of photons, how the heck does the color of photons delivering most cumulative power change whether I choose to describe that color based on its wavelength or it's frequency?
Is there an analogue to something like sound energy or is this quantum physics weirdness?
(These are semi rhetorical questions... I'm not expecting you to explain unless you really feel like it 馃榾)
So we can see the where this weirdness comes from when we look at the energy for a photon, E=hf=hc/位
When we integrate we sort of slice the function in fixed intervals, what i called above df and d位. So let's see what is the difference in energy when our frequency interval is, for example, 1000 Hz, and use a concrete example with 100 Hz and 1100 Hz. Then 螖E = E(1100 Hz) - E(100 Hz) = h路(1100 Hz - 100 Hz) = h路(1000 Hz) = 6.626脳10^-31 joules. You can check that this difference in energy will be the same if we had used any other frequencies as long as they had been 1000 Hz apart.
Now let's do the same with a fixed interval in wavelength. We'll use 1000 nm and start at 100 nm. Then 螖E = E(100 nm) - E(1100 nm) = hc路(1/(100 nm)-1/(1100 nm)) = 1.806脳10^-18 joules. This energy corresponds to a frequency interval of 2.725脳10^15 hertz. Now let's do one more step. 螖E = E(1100 nm) - E(2100 nm) = 8.599脳10^-20 joules, which corresponds to a frequency interval of 1.298脳10^14 hertz.
So the energy emitted in a fixed frequency interval is not comparable to the energy emitted in a wavelength interval. To account for this the very function that is being integrated has to be different, as in the end what's relevant is the result of the integral: the total energy radiated. This result has to be the same independent of the variable we use to integrate. That's why the peaks in frequency are different to those in wavelength: the peaks depend on the function, and the functions aren't the same.